Many people thinking of learning to play the guitar ask about nails. Should they be long or short? The short answer is that it depends:
The Fretting Hand
Guitarists of all styles agree that the fingernails on whichever hand you use to press down notes and chords (your left hand if you're right handed) should be kept short. Long nails can make it difficult to press notes down, leading to unwanted buzzes.
The Picking Hand
This is where it gets more complicated. Whether you play with or without nails depends a lot on style. If you want to play pop music or electric guitar, then chances are that you'll be playing with a pick. In this case, you don't need long nails on your picking hand. For certain rock guitar techniques such as "tapping", nails could potentially get in the way if they're too long, so short nails are generally preferable on the picking hand too.
What about classical, flamenco and fingerstyle guitar?
The majority of classical, flamenco and fingerstyle guitarists have long nails on their picking hand. The string is not plucked with flesh or nail alone, but with a combination of the two. Most of the tone production comes from the finger itself, with the nail just being "the icing on the cake", and helping to provide a bit of volume and definition.
To some extent, the nail length is a matter of personal preference. Some guitarists keep them barely long enough to protrude slightly over the end of the fingertip. Others prefer much longer nails.
What if I have brittle nails?
Guitarists with naturally weak nails sometimes use false nails, or use special products to strengthen the nails. Classical, flamenco and fingerstyle guitarist also shape their nails carefully with a nail buffer to achieve a very smooth edge. This ensures a good tone and also makes them less likely to break.
The type of nail shape you should cultivate depends partly on what kind of nails you have. Books have been written on the subject of nails. No, we're not joking!:
Can I play classical, flamenco or fingerstyle without nails?
Absolutely! Unless you're a planning on becoming a professional concert guitarist, then you can certainly play without nails if that suits you better. There are even a few excellent professional guitarists who play without nails, such as Virginia Luque. Historically, many guitarists - including the great Fernando Sor no less - played without nails because they believed it produced a warmer tone, and volume was considered less of a priority than it is today.
So in summary, whether you play with or without nails on your picking hand depends largely on what style of music you want to play. Most professional classical, flamenco and fingerstyle guitarists do cultivate nails, although the length will vary from person to person. However, nails are by no means essential, especially when you're still starting out. You can still be an excellent classical, flamenco or fingerstyle player without them. Indeed, having overly long nails in the early stages may even impede your progress, in the same way that a runner's technique might be impaired by only ever wearing thick-soled running shoes. As you progress, experiment with adding a bit of nail, starting with just a little and keep them well maintained. Again, think of them as "the icing on the cake".